# Properties

 Label 1014.4.b.c.337.1 Level $1014$ Weight $4$ Character 1014.337 Analytic conductor $59.828$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,4,Mod(337,1014)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1014.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$59.8279367458$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.337 Dual form 1014.4.b.c.337.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} -6.00000i q^{5} +6.00000i q^{6} +20.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} -6.00000i q^{5} +6.00000i q^{6} +20.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} -12.0000 q^{10} +24.0000i q^{11} +12.0000 q^{12} +40.0000 q^{14} +18.0000i q^{15} +16.0000 q^{16} +30.0000 q^{17} -18.0000i q^{18} +16.0000i q^{19} +24.0000i q^{20} -60.0000i q^{21} +48.0000 q^{22} +72.0000 q^{23} -24.0000i q^{24} +89.0000 q^{25} -27.0000 q^{27} -80.0000i q^{28} -282.000 q^{29} +36.0000 q^{30} -164.000i q^{31} -32.0000i q^{32} -72.0000i q^{33} -60.0000i q^{34} +120.000 q^{35} -36.0000 q^{36} +110.000i q^{37} +32.0000 q^{38} +48.0000 q^{40} +126.000i q^{41} -120.000 q^{42} -164.000 q^{43} -96.0000i q^{44} -54.0000i q^{45} -144.000i q^{46} -204.000i q^{47} -48.0000 q^{48} -57.0000 q^{49} -178.000i q^{50} -90.0000 q^{51} -738.000 q^{53} +54.0000i q^{54} +144.000 q^{55} -160.000 q^{56} -48.0000i q^{57} +564.000i q^{58} +120.000i q^{59} -72.0000i q^{60} +614.000 q^{61} -328.000 q^{62} +180.000i q^{63} -64.0000 q^{64} -144.000 q^{66} -848.000i q^{67} -120.000 q^{68} -216.000 q^{69} -240.000i q^{70} -132.000i q^{71} +72.0000i q^{72} +218.000i q^{73} +220.000 q^{74} -267.000 q^{75} -64.0000i q^{76} -480.000 q^{77} -1096.00 q^{79} -96.0000i q^{80} +81.0000 q^{81} +252.000 q^{82} -552.000i q^{83} +240.000i q^{84} -180.000i q^{85} +328.000i q^{86} +846.000 q^{87} -192.000 q^{88} +210.000i q^{89} -108.000 q^{90} -288.000 q^{92} +492.000i q^{93} -408.000 q^{94} +96.0000 q^{95} +96.0000i q^{96} +1726.00i q^{97} +114.000i q^{98} +216.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 $$2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} - 24 q^{10} + 24 q^{12} + 80 q^{14} + 32 q^{16} + 60 q^{17} + 96 q^{22} + 144 q^{23} + 178 q^{25} - 54 q^{27} - 564 q^{29} + 72 q^{30} + 240 q^{35} - 72 q^{36} + 64 q^{38} + 96 q^{40} - 240 q^{42} - 328 q^{43} - 96 q^{48} - 114 q^{49} - 180 q^{51} - 1476 q^{53} + 288 q^{55} - 320 q^{56} + 1228 q^{61} - 656 q^{62} - 128 q^{64} - 288 q^{66} - 240 q^{68} - 432 q^{69} + 440 q^{74} - 534 q^{75} - 960 q^{77} - 2192 q^{79} + 162 q^{81} + 504 q^{82} + 1692 q^{87} - 384 q^{88} - 216 q^{90} - 576 q^{92} - 816 q^{94} + 192 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 - 24 * q^10 + 24 * q^12 + 80 * q^14 + 32 * q^16 + 60 * q^17 + 96 * q^22 + 144 * q^23 + 178 * q^25 - 54 * q^27 - 564 * q^29 + 72 * q^30 + 240 * q^35 - 72 * q^36 + 64 * q^38 + 96 * q^40 - 240 * q^42 - 328 * q^43 - 96 * q^48 - 114 * q^49 - 180 * q^51 - 1476 * q^53 + 288 * q^55 - 320 * q^56 + 1228 * q^61 - 656 * q^62 - 128 * q^64 - 288 * q^66 - 240 * q^68 - 432 * q^69 + 440 * q^74 - 534 * q^75 - 960 * q^77 - 2192 * q^79 + 162 * q^81 + 504 * q^82 + 1692 * q^87 - 384 * q^88 - 216 * q^90 - 576 * q^92 - 816 * q^94 + 192 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 0.707107i
$$3$$ −3.00000 −0.577350
$$4$$ −4.00000 −0.500000
$$5$$ − 6.00000i − 0.536656i −0.963328 0.268328i $$-0.913529\pi$$
0.963328 0.268328i $$-0.0864711\pi$$
$$6$$ 6.00000i 0.408248i
$$7$$ 20.0000i 1.07990i 0.841698 + 0.539949i $$0.181557\pi$$
−0.841698 + 0.539949i $$0.818443\pi$$
$$8$$ 8.00000i 0.353553i
$$9$$ 9.00000 0.333333
$$10$$ −12.0000 −0.379473
$$11$$ 24.0000i 0.657843i 0.944357 + 0.328921i $$0.106685\pi$$
−0.944357 + 0.328921i $$0.893315\pi$$
$$12$$ 12.0000 0.288675
$$13$$ 0 0
$$14$$ 40.0000 0.763604
$$15$$ 18.0000i 0.309839i
$$16$$ 16.0000 0.250000
$$17$$ 30.0000 0.428004 0.214002 0.976833i $$-0.431350\pi$$
0.214002 + 0.976833i $$0.431350\pi$$
$$18$$ − 18.0000i − 0.235702i
$$19$$ 16.0000i 0.193192i 0.995324 + 0.0965961i $$0.0307955\pi$$
−0.995324 + 0.0965961i $$0.969204\pi$$
$$20$$ 24.0000i 0.268328i
$$21$$ − 60.0000i − 0.623480i
$$22$$ 48.0000 0.465165
$$23$$ 72.0000 0.652741 0.326370 0.945242i $$-0.394174\pi$$
0.326370 + 0.945242i $$0.394174\pi$$
$$24$$ − 24.0000i − 0.204124i
$$25$$ 89.0000 0.712000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ − 80.0000i − 0.539949i
$$29$$ −282.000 −1.80573 −0.902864 0.429927i $$-0.858539\pi$$
−0.902864 + 0.429927i $$0.858539\pi$$
$$30$$ 36.0000 0.219089
$$31$$ − 164.000i − 0.950170i −0.879940 0.475085i $$-0.842417\pi$$
0.879940 0.475085i $$-0.157583\pi$$
$$32$$ − 32.0000i − 0.176777i
$$33$$ − 72.0000i − 0.379806i
$$34$$ − 60.0000i − 0.302645i
$$35$$ 120.000 0.579534
$$36$$ −36.0000 −0.166667
$$37$$ 110.000i 0.488754i 0.969680 + 0.244377i $$0.0785834\pi$$
−0.969680 + 0.244377i $$0.921417\pi$$
$$38$$ 32.0000 0.136608
$$39$$ 0 0
$$40$$ 48.0000 0.189737
$$41$$ 126.000i 0.479949i 0.970779 + 0.239974i $$0.0771390\pi$$
−0.970779 + 0.239974i $$0.922861\pi$$
$$42$$ −120.000 −0.440867
$$43$$ −164.000 −0.581622 −0.290811 0.956780i $$-0.593925\pi$$
−0.290811 + 0.956780i $$0.593925\pi$$
$$44$$ − 96.0000i − 0.328921i
$$45$$ − 54.0000i − 0.178885i
$$46$$ − 144.000i − 0.461557i
$$47$$ − 204.000i − 0.633116i −0.948573 0.316558i $$-0.897473\pi$$
0.948573 0.316558i $$-0.102527\pi$$
$$48$$ −48.0000 −0.144338
$$49$$ −57.0000 −0.166181
$$50$$ − 178.000i − 0.503460i
$$51$$ −90.0000 −0.247108
$$52$$ 0 0
$$53$$ −738.000 −1.91268 −0.956341 0.292255i $$-0.905595\pi$$
−0.956341 + 0.292255i $$0.905595\pi$$
$$54$$ 54.0000i 0.136083i
$$55$$ 144.000 0.353036
$$56$$ −160.000 −0.381802
$$57$$ − 48.0000i − 0.111540i
$$58$$ 564.000i 1.27684i
$$59$$ 120.000i 0.264791i 0.991197 + 0.132396i $$0.0422669\pi$$
−0.991197 + 0.132396i $$0.957733\pi$$
$$60$$ − 72.0000i − 0.154919i
$$61$$ 614.000 1.28876 0.644382 0.764703i $$-0.277115\pi$$
0.644382 + 0.764703i $$0.277115\pi$$
$$62$$ −328.000 −0.671872
$$63$$ 180.000i 0.359966i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ −144.000 −0.268563
$$67$$ − 848.000i − 1.54626i −0.634245 0.773132i $$-0.718689\pi$$
0.634245 0.773132i $$-0.281311\pi$$
$$68$$ −120.000 −0.214002
$$69$$ −216.000 −0.376860
$$70$$ − 240.000i − 0.409793i
$$71$$ − 132.000i − 0.220641i −0.993896 0.110321i $$-0.964812\pi$$
0.993896 0.110321i $$-0.0351877\pi$$
$$72$$ 72.0000i 0.117851i
$$73$$ 218.000i 0.349520i 0.984611 + 0.174760i $$0.0559150\pi$$
−0.984611 + 0.174760i $$0.944085\pi$$
$$74$$ 220.000 0.345601
$$75$$ −267.000 −0.411073
$$76$$ − 64.0000i − 0.0965961i
$$77$$ −480.000 −0.710404
$$78$$ 0 0
$$79$$ −1096.00 −1.56088 −0.780441 0.625230i $$-0.785005\pi$$
−0.780441 + 0.625230i $$0.785005\pi$$
$$80$$ − 96.0000i − 0.134164i
$$81$$ 81.0000 0.111111
$$82$$ 252.000 0.339375
$$83$$ − 552.000i − 0.729998i −0.931008 0.364999i $$-0.881069\pi$$
0.931008 0.364999i $$-0.118931\pi$$
$$84$$ 240.000i 0.311740i
$$85$$ − 180.000i − 0.229691i
$$86$$ 328.000i 0.411269i
$$87$$ 846.000 1.04254
$$88$$ −192.000 −0.232583
$$89$$ 210.000i 0.250112i 0.992150 + 0.125056i $$0.0399110\pi$$
−0.992150 + 0.125056i $$0.960089\pi$$
$$90$$ −108.000 −0.126491
$$91$$ 0 0
$$92$$ −288.000 −0.326370
$$93$$ 492.000i 0.548581i
$$94$$ −408.000 −0.447681
$$95$$ 96.0000 0.103678
$$96$$ 96.0000i 0.102062i
$$97$$ 1726.00i 1.80669i 0.428917 + 0.903344i $$0.358895\pi$$
−0.428917 + 0.903344i $$0.641105\pi$$
$$98$$ 114.000i 0.117508i
$$99$$ 216.000i 0.219281i
$$100$$ −356.000 −0.356000
$$101$$ −798.000 −0.786178 −0.393089 0.919500i $$-0.628594\pi$$
−0.393089 + 0.919500i $$0.628594\pi$$
$$102$$ 180.000i 0.174732i
$$103$$ 520.000 0.497448 0.248724 0.968574i $$-0.419989\pi$$
0.248724 + 0.968574i $$0.419989\pi$$
$$104$$ 0 0
$$105$$ −360.000 −0.334594
$$106$$ 1476.00i 1.35247i
$$107$$ 12.0000 0.0108419 0.00542095 0.999985i $$-0.498274\pi$$
0.00542095 + 0.999985i $$0.498274\pi$$
$$108$$ 108.000 0.0962250
$$109$$ 1834.00i 1.61161i 0.592182 + 0.805804i $$0.298267\pi$$
−0.592182 + 0.805804i $$0.701733\pi$$
$$110$$ − 288.000i − 0.249634i
$$111$$ − 330.000i − 0.282182i
$$112$$ 320.000i 0.269975i
$$113$$ −366.000 −0.304694 −0.152347 0.988327i $$-0.548683\pi$$
−0.152347 + 0.988327i $$0.548683\pi$$
$$114$$ −96.0000 −0.0788704
$$115$$ − 432.000i − 0.350297i
$$116$$ 1128.00 0.902864
$$117$$ 0 0
$$118$$ 240.000 0.187236
$$119$$ 600.000i 0.462201i
$$120$$ −144.000 −0.109545
$$121$$ 755.000 0.567243
$$122$$ − 1228.00i − 0.911294i
$$123$$ − 378.000i − 0.277098i
$$124$$ 656.000i 0.475085i
$$125$$ − 1284.00i − 0.918756i
$$126$$ 360.000 0.254535
$$127$$ −2144.00 −1.49803 −0.749013 0.662556i $$-0.769472\pi$$
−0.749013 + 0.662556i $$0.769472\pi$$
$$128$$ 128.000i 0.0883883i
$$129$$ 492.000 0.335800
$$130$$ 0 0
$$131$$ −2748.00 −1.83278 −0.916389 0.400289i $$-0.868910\pi$$
−0.916389 + 0.400289i $$0.868910\pi$$
$$132$$ 288.000i 0.189903i
$$133$$ −320.000 −0.208628
$$134$$ −1696.00 −1.09337
$$135$$ 162.000i 0.103280i
$$136$$ 240.000i 0.151322i
$$137$$ 2754.00i 1.71745i 0.512440 + 0.858723i $$0.328742\pi$$
−0.512440 + 0.858723i $$0.671258\pi$$
$$138$$ 432.000i 0.266480i
$$139$$ 2252.00 1.37419 0.687094 0.726568i $$-0.258886\pi$$
0.687094 + 0.726568i $$0.258886\pi$$
$$140$$ −480.000 −0.289767
$$141$$ 612.000i 0.365530i
$$142$$ −264.000 −0.156017
$$143$$ 0 0
$$144$$ 144.000 0.0833333
$$145$$ 1692.00i 0.969055i
$$146$$ 436.000 0.247148
$$147$$ 171.000 0.0959445
$$148$$ − 440.000i − 0.244377i
$$149$$ 1770.00i 0.973182i 0.873630 + 0.486591i $$0.161760\pi$$
−0.873630 + 0.486591i $$0.838240\pi$$
$$150$$ 534.000i 0.290673i
$$151$$ − 988.000i − 0.532466i −0.963909 0.266233i $$-0.914221\pi$$
0.963909 0.266233i $$-0.0857790\pi$$
$$152$$ −128.000 −0.0683038
$$153$$ 270.000 0.142668
$$154$$ 960.000i 0.502331i
$$155$$ −984.000 −0.509915
$$156$$ 0 0
$$157$$ 326.000 0.165717 0.0828587 0.996561i $$-0.473595\pi$$
0.0828587 + 0.996561i $$0.473595\pi$$
$$158$$ 2192.00i 1.10371i
$$159$$ 2214.00 1.10429
$$160$$ −192.000 −0.0948683
$$161$$ 1440.00i 0.704894i
$$162$$ − 162.000i − 0.0785674i
$$163$$ 1496.00i 0.718870i 0.933170 + 0.359435i $$0.117031\pi$$
−0.933170 + 0.359435i $$0.882969\pi$$
$$164$$ − 504.000i − 0.239974i
$$165$$ −432.000 −0.203825
$$166$$ −1104.00 −0.516187
$$167$$ 1116.00i 0.517118i 0.965995 + 0.258559i $$0.0832476\pi$$
−0.965995 + 0.258559i $$0.916752\pi$$
$$168$$ 480.000 0.220433
$$169$$ 0 0
$$170$$ −360.000 −0.162416
$$171$$ 144.000i 0.0643974i
$$172$$ 656.000 0.290811
$$173$$ −4374.00 −1.92225 −0.961124 0.276116i $$-0.910953\pi$$
−0.961124 + 0.276116i $$0.910953\pi$$
$$174$$ − 1692.00i − 0.737185i
$$175$$ 1780.00i 0.768888i
$$176$$ 384.000i 0.164461i
$$177$$ − 360.000i − 0.152877i
$$178$$ 420.000 0.176856
$$179$$ −12.0000 −0.00501074 −0.00250537 0.999997i $$-0.500797\pi$$
−0.00250537 + 0.999997i $$0.500797\pi$$
$$180$$ 216.000i 0.0894427i
$$181$$ −4718.00 −1.93749 −0.968746 0.248053i $$-0.920209\pi$$
−0.968746 + 0.248053i $$0.920209\pi$$
$$182$$ 0 0
$$183$$ −1842.00 −0.744069
$$184$$ 576.000i 0.230779i
$$185$$ 660.000 0.262293
$$186$$ 984.000 0.387905
$$187$$ 720.000i 0.281559i
$$188$$ 816.000i 0.316558i
$$189$$ − 540.000i − 0.207827i
$$190$$ − 192.000i − 0.0733113i
$$191$$ −1368.00 −0.518246 −0.259123 0.965844i $$-0.583434\pi$$
−0.259123 + 0.965844i $$0.583434\pi$$
$$192$$ 192.000 0.0721688
$$193$$ − 3310.00i − 1.23450i −0.786766 0.617251i $$-0.788246\pi$$
0.786766 0.617251i $$-0.211754\pi$$
$$194$$ 3452.00 1.27752
$$195$$ 0 0
$$196$$ 228.000 0.0830904
$$197$$ − 3126.00i − 1.13055i −0.824903 0.565275i $$-0.808770\pi$$
0.824903 0.565275i $$-0.191230\pi$$
$$198$$ 432.000 0.155055
$$199$$ −4664.00 −1.66142 −0.830709 0.556707i $$-0.812065\pi$$
−0.830709 + 0.556707i $$0.812065\pi$$
$$200$$ 712.000i 0.251730i
$$201$$ 2544.00i 0.892736i
$$202$$ 1596.00i 0.555912i
$$203$$ − 5640.00i − 1.95000i
$$204$$ 360.000 0.123554
$$205$$ 756.000 0.257567
$$206$$ − 1040.00i − 0.351749i
$$207$$ 648.000 0.217580
$$208$$ 0 0
$$209$$ −384.000 −0.127090
$$210$$ 720.000i 0.236594i
$$211$$ −556.000 −0.181406 −0.0907029 0.995878i $$-0.528911\pi$$
−0.0907029 + 0.995878i $$0.528911\pi$$
$$212$$ 2952.00 0.956341
$$213$$ 396.000i 0.127387i
$$214$$ − 24.0000i − 0.00766638i
$$215$$ 984.000i 0.312131i
$$216$$ − 216.000i − 0.0680414i
$$217$$ 3280.00 1.02609
$$218$$ 3668.00 1.13958
$$219$$ − 654.000i − 0.201796i
$$220$$ −576.000 −0.176518
$$221$$ 0 0
$$222$$ −660.000 −0.199533
$$223$$ 268.000i 0.0804781i 0.999190 + 0.0402390i $$0.0128119\pi$$
−0.999190 + 0.0402390i $$0.987188\pi$$
$$224$$ 640.000 0.190901
$$225$$ 801.000 0.237333
$$226$$ 732.000i 0.215451i
$$227$$ − 1800.00i − 0.526300i −0.964755 0.263150i $$-0.915239\pi$$
0.964755 0.263150i $$-0.0847615\pi$$
$$228$$ 192.000i 0.0557698i
$$229$$ 2990.00i 0.862816i 0.902157 + 0.431408i $$0.141983\pi$$
−0.902157 + 0.431408i $$0.858017\pi$$
$$230$$ −864.000 −0.247698
$$231$$ 1440.00 0.410152
$$232$$ − 2256.00i − 0.638421i
$$233$$ −2826.00 −0.794581 −0.397291 0.917693i $$-0.630049\pi$$
−0.397291 + 0.917693i $$0.630049\pi$$
$$234$$ 0 0
$$235$$ −1224.00 −0.339766
$$236$$ − 480.000i − 0.132396i
$$237$$ 3288.00 0.901175
$$238$$ 1200.00 0.326825
$$239$$ 1812.00i 0.490412i 0.969471 + 0.245206i $$0.0788556\pi$$
−0.969471 + 0.245206i $$0.921144\pi$$
$$240$$ 288.000i 0.0774597i
$$241$$ − 1582.00i − 0.422845i −0.977395 0.211422i $$-0.932190\pi$$
0.977395 0.211422i $$-0.0678096\pi$$
$$242$$ − 1510.00i − 0.401101i
$$243$$ −243.000 −0.0641500
$$244$$ −2456.00 −0.644382
$$245$$ 342.000i 0.0891820i
$$246$$ −756.000 −0.195938
$$247$$ 0 0
$$248$$ 1312.00 0.335936
$$249$$ 1656.00i 0.421465i
$$250$$ −2568.00 −0.649658
$$251$$ −2148.00 −0.540162 −0.270081 0.962838i $$-0.587050\pi$$
−0.270081 + 0.962838i $$0.587050\pi$$
$$252$$ − 720.000i − 0.179983i
$$253$$ 1728.00i 0.429401i
$$254$$ 4288.00i 1.05926i
$$255$$ 540.000i 0.132612i
$$256$$ 256.000 0.0625000
$$257$$ 558.000 0.135436 0.0677181 0.997704i $$-0.478428\pi$$
0.0677181 + 0.997704i $$0.478428\pi$$
$$258$$ − 984.000i − 0.237446i
$$259$$ −2200.00 −0.527804
$$260$$ 0 0
$$261$$ −2538.00 −0.601909
$$262$$ 5496.00i 1.29597i
$$263$$ 2112.00 0.495177 0.247588 0.968865i $$-0.420362\pi$$
0.247588 + 0.968865i $$0.420362\pi$$
$$264$$ 576.000 0.134282
$$265$$ 4428.00i 1.02645i
$$266$$ 640.000i 0.147522i
$$267$$ − 630.000i − 0.144402i
$$268$$ 3392.00i 0.773132i
$$269$$ 5046.00 1.14372 0.571859 0.820352i $$-0.306223\pi$$
0.571859 + 0.820352i $$0.306223\pi$$
$$270$$ 324.000 0.0730297
$$271$$ − 3796.00i − 0.850888i −0.904985 0.425444i $$-0.860118\pi$$
0.904985 0.425444i $$-0.139882\pi$$
$$272$$ 480.000 0.107001
$$273$$ 0 0
$$274$$ 5508.00 1.21442
$$275$$ 2136.00i 0.468384i
$$276$$ 864.000 0.188430
$$277$$ −5582.00 −1.21079 −0.605397 0.795924i $$-0.706986\pi$$
−0.605397 + 0.795924i $$0.706986\pi$$
$$278$$ − 4504.00i − 0.971698i
$$279$$ − 1476.00i − 0.316723i
$$280$$ 960.000i 0.204896i
$$281$$ − 1950.00i − 0.413976i −0.978343 0.206988i $$-0.933634\pi$$
0.978343 0.206988i $$-0.0663661\pi$$
$$282$$ 1224.00 0.258469
$$283$$ 4732.00 0.993951 0.496976 0.867765i $$-0.334444\pi$$
0.496976 + 0.867765i $$0.334444\pi$$
$$284$$ 528.000i 0.110321i
$$285$$ −288.000 −0.0598584
$$286$$ 0 0
$$287$$ −2520.00 −0.518296
$$288$$ − 288.000i − 0.0589256i
$$289$$ −4013.00 −0.816813
$$290$$ 3384.00 0.685225
$$291$$ − 5178.00i − 1.04309i
$$292$$ − 872.000i − 0.174760i
$$293$$ 4998.00i 0.996540i 0.867022 + 0.498270i $$0.166031\pi$$
−0.867022 + 0.498270i $$0.833969\pi$$
$$294$$ − 342.000i − 0.0678430i
$$295$$ 720.000 0.142102
$$296$$ −880.000 −0.172801
$$297$$ − 648.000i − 0.126602i
$$298$$ 3540.00 0.688143
$$299$$ 0 0
$$300$$ 1068.00 0.205537
$$301$$ − 3280.00i − 0.628093i
$$302$$ −1976.00 −0.376510
$$303$$ 2394.00 0.453900
$$304$$ 256.000i 0.0482980i
$$305$$ − 3684.00i − 0.691624i
$$306$$ − 540.000i − 0.100882i
$$307$$ 6824.00i 1.26862i 0.773079 + 0.634310i $$0.218716\pi$$
−0.773079 + 0.634310i $$0.781284\pi$$
$$308$$ 1920.00 0.355202
$$309$$ −1560.00 −0.287202
$$310$$ 1968.00i 0.360564i
$$311$$ 8760.00 1.59722 0.798608 0.601852i $$-0.205570\pi$$
0.798608 + 0.601852i $$0.205570\pi$$
$$312$$ 0 0
$$313$$ 3962.00 0.715481 0.357740 0.933821i $$-0.383547\pi$$
0.357740 + 0.933821i $$0.383547\pi$$
$$314$$ − 652.000i − 0.117180i
$$315$$ 1080.00 0.193178
$$316$$ 4384.00 0.780441
$$317$$ − 7086.00i − 1.25549i −0.778420 0.627744i $$-0.783979\pi$$
0.778420 0.627744i $$-0.216021\pi$$
$$318$$ − 4428.00i − 0.780849i
$$319$$ − 6768.00i − 1.18788i
$$320$$ 384.000i 0.0670820i
$$321$$ −36.0000 −0.00625958
$$322$$ 2880.00 0.498435
$$323$$ 480.000i 0.0826870i
$$324$$ −324.000 −0.0555556
$$325$$ 0 0
$$326$$ 2992.00 0.508318
$$327$$ − 5502.00i − 0.930463i
$$328$$ −1008.00 −0.169687
$$329$$ 4080.00 0.683701
$$330$$ 864.000i 0.144126i
$$331$$ 9016.00i 1.49717i 0.663037 + 0.748586i $$0.269267\pi$$
−0.663037 + 0.748586i $$0.730733\pi$$
$$332$$ 2208.00i 0.364999i
$$333$$ 990.000i 0.162918i
$$334$$ 2232.00 0.365658
$$335$$ −5088.00 −0.829812
$$336$$ − 960.000i − 0.155870i
$$337$$ −2306.00 −0.372747 −0.186374 0.982479i $$-0.559673\pi$$
−0.186374 + 0.982479i $$0.559673\pi$$
$$338$$ 0 0
$$339$$ 1098.00 0.175915
$$340$$ 720.000i 0.114846i
$$341$$ 3936.00 0.625063
$$342$$ 288.000 0.0455358
$$343$$ 5720.00i 0.900440i
$$344$$ − 1312.00i − 0.205635i
$$345$$ 1296.00i 0.202244i
$$346$$ 8748.00i 1.35924i
$$347$$ −11076.0 −1.71352 −0.856759 0.515717i $$-0.827526\pi$$
−0.856759 + 0.515717i $$0.827526\pi$$
$$348$$ −3384.00 −0.521269
$$349$$ 2342.00i 0.359210i 0.983739 + 0.179605i $$0.0574820\pi$$
−0.983739 + 0.179605i $$0.942518\pi$$
$$350$$ 3560.00 0.543686
$$351$$ 0 0
$$352$$ 768.000 0.116291
$$353$$ − 4650.00i − 0.701118i −0.936541 0.350559i $$-0.885992\pi$$
0.936541 0.350559i $$-0.114008\pi$$
$$354$$ −720.000 −0.108100
$$355$$ −792.000 −0.118408
$$356$$ − 840.000i − 0.125056i
$$357$$ − 1800.00i − 0.266852i
$$358$$ 24.0000i 0.00354313i
$$359$$ − 11268.0i − 1.65655i −0.560320 0.828276i $$-0.689322\pi$$
0.560320 0.828276i $$-0.310678\pi$$
$$360$$ 432.000 0.0632456
$$361$$ 6603.00 0.962677
$$362$$ 9436.00i 1.37001i
$$363$$ −2265.00 −0.327498
$$364$$ 0 0
$$365$$ 1308.00 0.187572
$$366$$ 3684.00i 0.526136i
$$367$$ −7288.00 −1.03660 −0.518298 0.855200i $$-0.673434\pi$$
−0.518298 + 0.855200i $$0.673434\pi$$
$$368$$ 1152.00 0.163185
$$369$$ 1134.00i 0.159983i
$$370$$ − 1320.00i − 0.185469i
$$371$$ − 14760.0i − 2.06550i
$$372$$ − 1968.00i − 0.274290i
$$373$$ −9970.00 −1.38399 −0.691993 0.721904i $$-0.743267\pi$$
−0.691993 + 0.721904i $$0.743267\pi$$
$$374$$ 1440.00 0.199093
$$375$$ 3852.00i 0.530444i
$$376$$ 1632.00 0.223840
$$377$$ 0 0
$$378$$ −1080.00 −0.146956
$$379$$ − 13448.0i − 1.82263i −0.411708 0.911316i $$-0.635068\pi$$
0.411708 0.911316i $$-0.364932\pi$$
$$380$$ −384.000 −0.0518389
$$381$$ 6432.00 0.864885
$$382$$ 2736.00i 0.366455i
$$383$$ − 11820.0i − 1.57696i −0.615064 0.788478i $$-0.710870\pi$$
0.615064 0.788478i $$-0.289130\pi$$
$$384$$ − 384.000i − 0.0510310i
$$385$$ 2880.00i 0.381243i
$$386$$ −6620.00 −0.872925
$$387$$ −1476.00 −0.193874
$$388$$ − 6904.00i − 0.903344i
$$389$$ −174.000 −0.0226790 −0.0113395 0.999936i $$-0.503610\pi$$
−0.0113395 + 0.999936i $$0.503610\pi$$
$$390$$ 0 0
$$391$$ 2160.00 0.279376
$$392$$ − 456.000i − 0.0587538i
$$393$$ 8244.00 1.05815
$$394$$ −6252.00 −0.799419
$$395$$ 6576.00i 0.837657i
$$396$$ − 864.000i − 0.109640i
$$397$$ − 2986.00i − 0.377489i −0.982026 0.188744i $$-0.939558\pi$$
0.982026 0.188744i $$-0.0604418\pi$$
$$398$$ 9328.00i 1.17480i
$$399$$ 960.000 0.120451
$$400$$ 1424.00 0.178000
$$401$$ − 10566.0i − 1.31581i −0.753100 0.657906i $$-0.771442\pi$$
0.753100 0.657906i $$-0.228558\pi$$
$$402$$ 5088.00 0.631260
$$403$$ 0 0
$$404$$ 3192.00 0.393089
$$405$$ − 486.000i − 0.0596285i
$$406$$ −11280.0 −1.37886
$$407$$ −2640.00 −0.321523
$$408$$ − 720.000i − 0.0873660i
$$409$$ 7270.00i 0.878920i 0.898262 + 0.439460i $$0.144830\pi$$
−0.898262 + 0.439460i $$0.855170\pi$$
$$410$$ − 1512.00i − 0.182128i
$$411$$ − 8262.00i − 0.991568i
$$412$$ −2080.00 −0.248724
$$413$$ −2400.00 −0.285947
$$414$$ − 1296.00i − 0.153852i
$$415$$ −3312.00 −0.391758
$$416$$ 0 0
$$417$$ −6756.00 −0.793388
$$418$$ 768.000i 0.0898663i
$$419$$ −7308.00 −0.852074 −0.426037 0.904706i $$-0.640091\pi$$
−0.426037 + 0.904706i $$0.640091\pi$$
$$420$$ 1440.00 0.167297
$$421$$ 5938.00i 0.687412i 0.939077 + 0.343706i $$0.111682\pi$$
−0.939077 + 0.343706i $$0.888318\pi$$
$$422$$ 1112.00i 0.128273i
$$423$$ − 1836.00i − 0.211039i
$$424$$ − 5904.00i − 0.676235i
$$425$$ 2670.00 0.304739
$$426$$ 792.000 0.0900764
$$427$$ 12280.0i 1.39174i
$$428$$ −48.0000 −0.00542095
$$429$$ 0 0
$$430$$ 1968.00 0.220710
$$431$$ − 11532.0i − 1.28881i −0.764685 0.644405i $$-0.777105\pi$$
0.764685 0.644405i $$-0.222895\pi$$
$$432$$ −432.000 −0.0481125
$$433$$ 718.000 0.0796879 0.0398440 0.999206i $$-0.487314\pi$$
0.0398440 + 0.999206i $$0.487314\pi$$
$$434$$ − 6560.00i − 0.725553i
$$435$$ − 5076.00i − 0.559484i
$$436$$ − 7336.00i − 0.805804i
$$437$$ 1152.00i 0.126104i
$$438$$ −1308.00 −0.142691
$$439$$ −8984.00 −0.976726 −0.488363 0.872640i $$-0.662406\pi$$
−0.488363 + 0.872640i $$0.662406\pi$$
$$440$$ 1152.00i 0.124817i
$$441$$ −513.000 −0.0553936
$$442$$ 0 0
$$443$$ 2604.00 0.279277 0.139639 0.990203i $$-0.455406\pi$$
0.139639 + 0.990203i $$0.455406\pi$$
$$444$$ 1320.00i 0.141091i
$$445$$ 1260.00 0.134224
$$446$$ 536.000 0.0569066
$$447$$ − 5310.00i − 0.561867i
$$448$$ − 1280.00i − 0.134987i
$$449$$ − 13206.0i − 1.38804i −0.719956 0.694020i $$-0.755838\pi$$
0.719956 0.694020i $$-0.244162\pi$$
$$450$$ − 1602.00i − 0.167820i
$$451$$ −3024.00 −0.315731
$$452$$ 1464.00 0.152347
$$453$$ 2964.00i 0.307419i
$$454$$ −3600.00 −0.372151
$$455$$ 0 0
$$456$$ 384.000 0.0394352
$$457$$ − 8426.00i − 0.862476i −0.902238 0.431238i $$-0.858077\pi$$
0.902238 0.431238i $$-0.141923\pi$$
$$458$$ 5980.00 0.610103
$$459$$ −810.000 −0.0823694
$$460$$ 1728.00i 0.175149i
$$461$$ − 16686.0i − 1.68578i −0.538086 0.842890i $$-0.680852\pi$$
0.538086 0.842890i $$-0.319148\pi$$
$$462$$ − 2880.00i − 0.290021i
$$463$$ 15932.0i 1.59919i 0.600543 + 0.799593i $$0.294951\pi$$
−0.600543 + 0.799593i $$0.705049\pi$$
$$464$$ −4512.00 −0.451432
$$465$$ 2952.00 0.294399
$$466$$ 5652.00i 0.561854i
$$467$$ −18540.0 −1.83711 −0.918553 0.395297i $$-0.870642\pi$$
−0.918553 + 0.395297i $$0.870642\pi$$
$$468$$ 0 0
$$469$$ 16960.0 1.66981
$$470$$ 2448.00i 0.240251i
$$471$$ −978.000 −0.0956770
$$472$$ −960.000 −0.0936178
$$473$$ − 3936.00i − 0.382616i
$$474$$ − 6576.00i − 0.637227i
$$475$$ 1424.00i 0.137553i
$$476$$ − 2400.00i − 0.231100i
$$477$$ −6642.00 −0.637560
$$478$$ 3624.00 0.346774
$$479$$ 6180.00i 0.589502i 0.955574 + 0.294751i $$0.0952367\pi$$
−0.955574 + 0.294751i $$0.904763\pi$$
$$480$$ 576.000 0.0547723
$$481$$ 0 0
$$482$$ −3164.00 −0.298996
$$483$$ − 4320.00i − 0.406971i
$$484$$ −3020.00 −0.283621
$$485$$ 10356.0 0.969571
$$486$$ 486.000i 0.0453609i
$$487$$ − 11756.0i − 1.09387i −0.837175 0.546936i $$-0.815794\pi$$
0.837175 0.546936i $$-0.184206\pi$$
$$488$$ 4912.00i 0.455647i
$$489$$ − 4488.00i − 0.415040i
$$490$$ 684.000 0.0630612
$$491$$ −1908.00 −0.175370 −0.0876852 0.996148i $$-0.527947\pi$$
−0.0876852 + 0.996148i $$0.527947\pi$$
$$492$$ 1512.00i 0.138549i
$$493$$ −8460.00 −0.772858
$$494$$ 0 0
$$495$$ 1296.00 0.117679
$$496$$ − 2624.00i − 0.237542i
$$497$$ 2640.00 0.238270
$$498$$ 3312.00 0.298021
$$499$$ 8944.00i 0.802382i 0.915995 + 0.401191i $$0.131404\pi$$
−0.915995 + 0.401191i $$0.868596\pi$$
$$500$$ 5136.00i 0.459378i
$$501$$ − 3348.00i − 0.298558i
$$502$$ 4296.00i 0.381952i
$$503$$ −6528.00 −0.578666 −0.289333 0.957228i $$-0.593434\pi$$
−0.289333 + 0.957228i $$0.593434\pi$$
$$504$$ −1440.00 −0.127267
$$505$$ 4788.00i 0.421907i
$$506$$ 3456.00 0.303632
$$507$$ 0 0
$$508$$ 8576.00 0.749013
$$509$$ 12114.0i 1.05490i 0.849586 + 0.527450i $$0.176852\pi$$
−0.849586 + 0.527450i $$0.823148\pi$$
$$510$$ 1080.00 0.0937710
$$511$$ −4360.00 −0.377446
$$512$$ − 512.000i − 0.0441942i
$$513$$ − 432.000i − 0.0371799i
$$514$$ − 1116.00i − 0.0957678i
$$515$$ − 3120.00i − 0.266958i
$$516$$ −1968.00 −0.167900
$$517$$ 4896.00 0.416491
$$518$$ 4400.00i 0.373214i
$$519$$ 13122.0 1.10981
$$520$$ 0 0
$$521$$ −14310.0 −1.20333 −0.601663 0.798750i $$-0.705495\pi$$
−0.601663 + 0.798750i $$0.705495\pi$$
$$522$$ 5076.00i 0.425614i
$$523$$ −18340.0 −1.53337 −0.766685 0.642024i $$-0.778095\pi$$
−0.766685 + 0.642024i $$0.778095\pi$$
$$524$$ 10992.0 0.916389
$$525$$ − 5340.00i − 0.443918i
$$526$$ − 4224.00i − 0.350143i
$$527$$ − 4920.00i − 0.406677i
$$528$$ − 1152.00i − 0.0949514i
$$529$$ −6983.00 −0.573929
$$530$$ 8856.00 0.725811
$$531$$ 1080.00i 0.0882637i
$$532$$ 1280.00 0.104314
$$533$$ 0 0
$$534$$ −1260.00 −0.102108
$$535$$ − 72.0000i − 0.00581838i
$$536$$ 6784.00 0.546687
$$537$$ 36.0000 0.00289295
$$538$$ − 10092.0i − 0.808731i
$$539$$ − 1368.00i − 0.109321i
$$540$$ − 648.000i − 0.0516398i
$$541$$ 9254.00i 0.735417i 0.929941 + 0.367708i $$0.119858\pi$$
−0.929941 + 0.367708i $$0.880142\pi$$
$$542$$ −7592.00 −0.601668
$$543$$ 14154.0 1.11861
$$544$$ − 960.000i − 0.0756611i
$$545$$ 11004.0 0.864880
$$546$$ 0 0
$$547$$ 17444.0 1.36353 0.681766 0.731571i $$-0.261212\pi$$
0.681766 + 0.731571i $$0.261212\pi$$
$$548$$ − 11016.0i − 0.858723i
$$549$$ 5526.00 0.429588
$$550$$ 4272.00 0.331198
$$551$$ − 4512.00i − 0.348852i
$$552$$ − 1728.00i − 0.133240i
$$553$$ − 21920.0i − 1.68559i
$$554$$ 11164.0i 0.856160i
$$555$$ −1980.00 −0.151435
$$556$$ −9008.00 −0.687094
$$557$$ − 3714.00i − 0.282526i −0.989972 0.141263i $$-0.954884\pi$$
0.989972 0.141263i $$-0.0451164\pi$$
$$558$$ −2952.00 −0.223957
$$559$$ 0 0
$$560$$ 1920.00 0.144884
$$561$$ − 2160.00i − 0.162558i
$$562$$ −3900.00 −0.292725
$$563$$ 13812.0 1.03394 0.516968 0.856004i $$-0.327060\pi$$
0.516968 + 0.856004i $$0.327060\pi$$
$$564$$ − 2448.00i − 0.182765i
$$565$$ 2196.00i 0.163516i
$$566$$ − 9464.00i − 0.702830i
$$567$$ 1620.00i 0.119989i
$$568$$ 1056.00 0.0780084
$$569$$ 15942.0 1.17456 0.587279 0.809385i $$-0.300199\pi$$
0.587279 + 0.809385i $$0.300199\pi$$
$$570$$ 576.000i 0.0423263i
$$571$$ −1604.00 −0.117557 −0.0587787 0.998271i $$-0.518721\pi$$
−0.0587787 + 0.998271i $$0.518721\pi$$
$$572$$ 0 0
$$573$$ 4104.00 0.299210
$$574$$ 5040.00i 0.366490i
$$575$$ 6408.00 0.464751
$$576$$ −576.000 −0.0416667
$$577$$ 10654.0i 0.768686i 0.923190 + 0.384343i $$0.125572\pi$$
−0.923190 + 0.384343i $$0.874428\pi$$
$$578$$ 8026.00i 0.577574i
$$579$$ 9930.00i 0.712740i
$$580$$ − 6768.00i − 0.484527i
$$581$$ 11040.0 0.788324
$$582$$ −10356.0 −0.737577
$$583$$ − 17712.0i − 1.25824i
$$584$$ −1744.00 −0.123574
$$585$$ 0 0
$$586$$ 9996.00 0.704660
$$587$$ 9984.00i 0.702017i 0.936372 + 0.351008i $$0.114161\pi$$
−0.936372 + 0.351008i $$0.885839\pi$$
$$588$$ −684.000 −0.0479723
$$589$$ 2624.00 0.183565
$$590$$ − 1440.00i − 0.100481i
$$591$$ 9378.00i 0.652723i
$$592$$ 1760.00i 0.122188i
$$593$$ 12618.0i 0.873793i 0.899512 + 0.436896i $$0.143922\pi$$
−0.899512 + 0.436896i $$0.856078\pi$$
$$594$$ −1296.00 −0.0895211
$$595$$ 3600.00 0.248043
$$596$$ − 7080.00i − 0.486591i
$$597$$ 13992.0 0.959220
$$598$$ 0 0
$$599$$ 11184.0 0.762881 0.381441 0.924393i $$-0.375428\pi$$
0.381441 + 0.924393i $$0.375428\pi$$
$$600$$ − 2136.00i − 0.145336i
$$601$$ 2810.00 0.190719 0.0953596 0.995443i $$-0.469600\pi$$
0.0953596 + 0.995443i $$0.469600\pi$$
$$602$$ −6560.00 −0.444129
$$603$$ − 7632.00i − 0.515421i
$$604$$ 3952.00i 0.266233i
$$605$$ − 4530.00i − 0.304414i
$$606$$ − 4788.00i − 0.320956i
$$607$$ 1064.00 0.0711473 0.0355737 0.999367i $$-0.488674\pi$$
0.0355737 + 0.999367i $$0.488674\pi$$
$$608$$ 512.000 0.0341519
$$609$$ 16920.0i 1.12583i
$$610$$ −7368.00 −0.489052
$$611$$ 0 0
$$612$$ −1080.00 −0.0713340
$$613$$ 20914.0i 1.37799i 0.724766 + 0.688996i $$0.241948\pi$$
−0.724766 + 0.688996i $$0.758052\pi$$
$$614$$ 13648.0 0.897050
$$615$$ −2268.00 −0.148707
$$616$$ − 3840.00i − 0.251166i
$$617$$ − 9714.00i − 0.633826i −0.948455 0.316913i $$-0.897354\pi$$
0.948455 0.316913i $$-0.102646\pi$$
$$618$$ 3120.00i 0.203082i
$$619$$ − 14848.0i − 0.964122i −0.876138 0.482061i $$-0.839888\pi$$
0.876138 0.482061i $$-0.160112\pi$$
$$620$$ 3936.00 0.254957
$$621$$ −1944.00 −0.125620
$$622$$ − 17520.0i − 1.12940i
$$623$$ −4200.00 −0.270095
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ − 7924.00i − 0.505921i
$$627$$ 1152.00 0.0733755
$$628$$ −1304.00 −0.0828587
$$629$$ 3300.00i 0.209189i
$$630$$ − 2160.00i − 0.136598i
$$631$$ 19172.0i 1.20955i 0.796397 + 0.604774i $$0.206737\pi$$
−0.796397 + 0.604774i $$0.793263\pi$$
$$632$$ − 8768.00i − 0.551855i
$$633$$ 1668.00 0.104735
$$634$$ −14172.0 −0.887763
$$635$$ 12864.0i 0.803925i
$$636$$ −8856.00 −0.552143
$$637$$ 0 0
$$638$$ −13536.0 −0.839961
$$639$$ − 1188.00i − 0.0735470i
$$640$$ 768.000 0.0474342
$$641$$ 11502.0 0.708739 0.354369 0.935105i $$-0.384696\pi$$
0.354369 + 0.935105i $$0.384696\pi$$
$$642$$ 72.0000i 0.00442619i
$$643$$ 15568.0i 0.954809i 0.878684 + 0.477404i $$0.158422\pi$$
−0.878684 + 0.477404i $$0.841578\pi$$
$$644$$ − 5760.00i − 0.352447i
$$645$$ − 2952.00i − 0.180209i
$$646$$ 960.000 0.0584686
$$647$$ −1128.00 −0.0685414 −0.0342707 0.999413i $$-0.510911\pi$$
−0.0342707 + 0.999413i $$0.510911\pi$$
$$648$$ 648.000i 0.0392837i
$$649$$ −2880.00 −0.174191
$$650$$ 0 0
$$651$$ −9840.00 −0.592412
$$652$$ − 5984.00i − 0.359435i
$$653$$ 8118.00 0.486496 0.243248 0.969964i $$-0.421787\pi$$
0.243248 + 0.969964i $$0.421787\pi$$
$$654$$ −11004.0 −0.657936
$$655$$ 16488.0i 0.983572i
$$656$$ 2016.00i 0.119987i
$$657$$ 1962.00i 0.116507i
$$658$$ − 8160.00i − 0.483450i
$$659$$ 13572.0 0.802261 0.401131 0.916021i $$-0.368617\pi$$
0.401131 + 0.916021i $$0.368617\pi$$
$$660$$ 1728.00 0.101913
$$661$$ − 13138.0i − 0.773085i −0.922272 0.386542i $$-0.873669\pi$$
0.922272 0.386542i $$-0.126331\pi$$
$$662$$ 18032.0 1.05866
$$663$$ 0 0
$$664$$ 4416.00 0.258093
$$665$$ 1920.00i 0.111962i
$$666$$ 1980.00 0.115200
$$667$$ −20304.0 −1.17867
$$668$$ − 4464.00i − 0.258559i
$$669$$ − 804.000i − 0.0464640i
$$670$$ 10176.0i 0.586766i
$$671$$ 14736.0i 0.847805i
$$672$$ −1920.00 −0.110217
$$673$$ 718.000 0.0411246 0.0205623 0.999789i $$-0.493454\pi$$
0.0205623 + 0.999789i $$0.493454\pi$$
$$674$$ 4612.00i 0.263572i
$$675$$ −2403.00 −0.137024
$$676$$ 0 0
$$677$$ −2994.00 −0.169969 −0.0849843 0.996382i $$-0.527084\pi$$
−0.0849843 + 0.996382i $$0.527084\pi$$
$$678$$ − 2196.00i − 0.124391i
$$679$$ −34520.0 −1.95104
$$680$$ 1440.00 0.0812081
$$681$$ 5400.00i 0.303860i
$$682$$ − 7872.00i − 0.441986i
$$683$$ 27384.0i 1.53414i 0.641562 + 0.767071i $$0.278287\pi$$
−0.641562 + 0.767071i $$0.721713\pi$$
$$684$$ − 576.000i − 0.0321987i
$$685$$ 16524.0 0.921678
$$686$$ 11440.0 0.636707
$$687$$ − 8970.00i − 0.498147i
$$688$$ −2624.00 −0.145406
$$689$$ 0 0
$$690$$ 2592.00 0.143008
$$691$$ − 27632.0i − 1.52123i −0.649202 0.760616i $$-0.724897\pi$$
0.649202 0.760616i $$-0.275103\pi$$
$$692$$ 17496.0 0.961124
$$693$$ −4320.00 −0.236801
$$694$$ 22152.0i 1.21164i
$$695$$ − 13512.0i − 0.737467i
$$696$$ 6768.00i 0.368592i
$$697$$ 3780.00i 0.205420i
$$698$$ 4684.00 0.254000
$$699$$ 8478.00 0.458752
$$700$$ − 7120.00i − 0.384444i
$$701$$ −19062.0 −1.02705 −0.513525 0.858075i $$-0.671661\pi$$
−0.513525 + 0.858075i $$0.671661\pi$$
$$702$$ 0 0
$$703$$ −1760.00 −0.0944234
$$704$$ − 1536.00i − 0.0822304i
$$705$$ 3672.00 0.196164
$$706$$ −9300.00 −0.495765
$$707$$ − 15960.0i − 0.848992i
$$708$$ 1440.00i 0.0764386i
$$709$$ 3854.00i 0.204147i 0.994777 + 0.102073i $$0.0325476\pi$$
−0.994777 + 0.102073i $$0.967452\pi$$
$$710$$ 1584.00i 0.0837274i
$$711$$ −9864.00 −0.520294
$$712$$ −1680.00 −0.0884279
$$713$$ − 11808.0i − 0.620215i
$$714$$ −3600.00 −0.188693
$$715$$ 0 0
$$716$$ 48.0000 0.00250537
$$717$$ − 5436.00i − 0.283140i
$$718$$ −22536.0 −1.17136
$$719$$ −20976.0 −1.08800 −0.544001 0.839085i $$-0.683091\pi$$
−0.544001 + 0.839085i $$0.683091\pi$$
$$720$$ − 864.000i − 0.0447214i
$$721$$ 10400.0i 0.537193i
$$722$$ − 13206.0i − 0.680715i
$$723$$ 4746.00i 0.244130i
$$724$$ 18872.0 0.968746
$$725$$ −25098.0 −1.28568
$$726$$ 4530.00i 0.231576i
$$727$$ 29464.0 1.50311 0.751554 0.659672i $$-0.229305\pi$$
0.751554 + 0.659672i $$0.229305\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ − 2616.00i − 0.132634i
$$731$$ −4920.00 −0.248937
$$732$$ 7368.00 0.372034
$$733$$ 2698.00i 0.135952i 0.997687 + 0.0679761i $$0.0216542\pi$$
−0.997687 + 0.0679761i $$0.978346\pi$$
$$734$$ 14576.0i 0.732984i
$$735$$ − 1026.00i − 0.0514892i
$$736$$ − 2304.00i − 0.115389i
$$737$$ 20352.0 1.01720
$$738$$ 2268.00 0.113125
$$739$$ 632.000i 0.0314594i 0.999876 + 0.0157297i $$0.00500713\pi$$
−0.999876 + 0.0157297i $$0.994993\pi$$
$$740$$ −2640.00 −0.131146
$$741$$ 0 0
$$742$$ −29520.0 −1.46053
$$743$$ 20844.0i 1.02920i 0.857432 + 0.514598i $$0.172059\pi$$
−0.857432 + 0.514598i $$0.827941\pi$$
$$744$$ −3936.00 −0.193953
$$745$$ 10620.0 0.522264
$$746$$ 19940.0i 0.978626i
$$747$$ − 4968.00i − 0.243333i
$$748$$ − 2880.00i − 0.140780i
$$749$$ 240.000i 0.0117082i
$$750$$ 7704.00 0.375080
$$751$$ −272.000 −0.0132163 −0.00660814 0.999978i $$-0.502103\pi$$
−0.00660814 + 0.999978i $$0.502103\pi$$
$$752$$ − 3264.00i − 0.158279i
$$753$$ 6444.00 0.311862
$$754$$ 0 0
$$755$$ −5928.00 −0.285751
$$756$$ 2160.00i 0.103913i
$$757$$ 37550.0 1.80288 0.901439 0.432907i $$-0.142512\pi$$
0.901439 + 0.432907i $$0.142512\pi$$
$$758$$ −26896.0 −1.28880
$$759$$ − 5184.00i − 0.247915i
$$760$$ 768.000i 0.0366556i
$$761$$ 33330.0i 1.58766i 0.608138 + 0.793832i $$0.291917\pi$$
−0.608138 + 0.793832i $$0.708083\pi$$
$$762$$ − 12864.0i − 0.611566i
$$763$$ −36680.0 −1.74037
$$764$$ 5472.00 0.259123
$$765$$ − 1620.00i − 0.0765637i
$$766$$ −23640.0 −1.11508
$$767$$ 0 0
$$768$$ −768.000 −0.0360844
$$769$$ 15406.0i 0.722438i 0.932481 + 0.361219i $$0.117639\pi$$
−0.932481 + 0.361219i $$0.882361\pi$$
$$770$$ 5760.00 0.269579
$$771$$ −1674.00 −0.0781941
$$772$$ 13240.0i 0.617251i
$$773$$ 29514.0i 1.37328i 0.726998 + 0.686640i $$0.240915\pi$$
−0.726998 + 0.686640i $$0.759085\pi$$
$$774$$ 2952.00i 0.137090i
$$775$$ − 14596.0i − 0.676521i
$$776$$ −13808.0 −0.638761
$$777$$ 6600.00 0.304728
$$778$$ 348.000i 0.0160365i
$$779$$ −2016.00 −0.0927223
$$780$$ 0 0
$$781$$ 3168.00 0.145147
$$782$$ − 4320.00i − 0.197548i
$$783$$ 7614.00 0.347512
$$784$$ −912.000 −0.0415452
$$785$$ − 1956.00i − 0.0889333i
$$786$$ − 16488.0i − 0.748228i
$$787$$ 33176.0i 1.50266i 0.659924 + 0.751332i $$0.270588\pi$$
−0.659924 + 0.751332i $$0.729412\pi$$
$$788$$ 12504.0i 0.565275i
$$789$$ −6336.00 −0.285890
$$790$$ 13152.0 0.592313
$$791$$ − 7320.00i − 0.329038i
$$792$$ −1728.00 −0.0775275
$$793$$ 0 0
$$794$$ −5972.00 −0.266925
$$795$$ − 13284.0i − 0.592623i
$$796$$ 18656.0 0.830709
$$797$$ 16746.0 0.744258 0.372129 0.928181i $$-0.378628\pi$$
0.372129 + 0.928181i $$0.378628\pi$$
$$798$$ − 1920.00i − 0.0851720i
$$799$$ − 6120.00i − 0.270976i
$$800$$ − 2848.00i − 0.125865i
$$801$$ 1890.00i 0.0833706i
$$802$$ −21132.0 −0.930420
$$803$$ −5232.00 −0.229929
$$804$$ − 10176.0i − 0.446368i
$$805$$ 8640.00 0.378286
$$806$$ 0 0
$$807$$ −15138.0 −0.660326
$$808$$ − 6384.00i − 0.277956i
$$809$$ −15846.0 −0.688647 −0.344324 0.938851i $$-0.611892\pi$$
−0.344324 + 0.938851i $$0.611892\pi$$
$$810$$ −972.000 −0.0421637
$$811$$ − 22952.0i − 0.993778i −0.867814 0.496889i $$-0.834476\pi$$
0.867814 0.496889i $$-0.165524\pi$$
$$812$$ 22560.0i 0.975001i
$$813$$ 11388.0i 0.491260i
$$814$$ 5280.00i 0.227351i
$$815$$ 8976.00 0.385786
$$816$$ −1440.00 −0.0617771
$$817$$ − 2624.00i − 0.112365i
$$818$$ 14540.0 0.621490
$$819$$ 0 0
$$820$$ −3024.00 −0.128784
$$821$$ 37146.0i 1.57906i 0.613715 + 0.789528i $$0.289674\pi$$
−0.613715 + 0.789528i $$0.710326\pi$$
$$822$$ −16524.0 −0.701144
$$823$$ 9592.00 0.406265 0.203133 0.979151i $$-0.434888\pi$$
0.203133 + 0.979151i $$0.434888\pi$$
$$824$$ 4160.00i 0.175874i
$$825$$ − 6408.00i − 0.270422i
$$826$$ 4800.00i 0.202195i
$$827$$ − 39960.0i − 1.68022i −0.542413 0.840112i $$-0.682489\pi$$
0.542413 0.840112i $$-0.317511\pi$$
$$828$$ −2592.00 −0.108790
$$829$$ 3706.00 0.155265 0.0776325 0.996982i $$-0.475264\pi$$
0.0776325 + 0.996982i $$0.475264\pi$$
$$830$$ 6624.00i 0.277015i
$$831$$ 16746.0 0.699052
$$832$$ 0 0
$$833$$ −1710.00 −0.0711260
$$834$$ 13512.0i 0.561010i
$$835$$ 6696.00 0.277515
$$836$$ 1536.00 0.0635451
$$837$$ 4428.00i 0.182860i
$$838$$ 14616.0i 0.602508i
$$839$$ 9756.00i 0.401448i 0.979648 + 0.200724i $$0.0643294\pi$$
−0.979648 + 0.200724i $$0.935671\pi$$
$$840$$ − 2880.00i − 0.118297i
$$841$$ 55135.0 2.26065
$$842$$ 11876.0 0.486074
$$843$$ 5850.00i 0.239009i
$$844$$ 2224.00 0.0907029
$$845$$ 0 0
$$846$$ −3672.00 −0.149227
$$847$$ 15100.0i 0.612565i
$$848$$ −11808.0 −0.478170
$$849$$ −14196.0 −0.573858
$$850$$ − 5340.00i − 0.215483i
$$851$$ 7920.00i 0.319029i
$$852$$ − 1584.00i − 0.0636936i
$$853$$ 11342.0i 0.455267i 0.973747 + 0.227633i $$0.0730988\pi$$
−0.973747 + 0.227633i $$0.926901\pi$$
$$854$$ 24560.0 0.984105
$$855$$ 864.000 0.0345593
$$856$$ 96.0000i 0.00383319i
$$857$$ 16134.0 0.643089 0.321544 0.946895i $$-0.395798\pi$$
0.321544 + 0.946895i $$0.395798\pi$$
$$858$$ 0 0
$$859$$ −20932.0 −0.831421 −0.415710 0.909497i $$-0.636467\pi$$
−0.415710 + 0.909497i $$0.636467\pi$$
$$860$$ − 3936.00i − 0.156066i
$$861$$ 7560.00 0.299238
$$862$$ −23064.0 −0.911326
$$863$$ − 10044.0i − 0.396178i −0.980184 0.198089i $$-0.936526\pi$$
0.980184 0.198089i $$-0.0634735\pi$$
$$864$$ 864.000i 0.0340207i
$$865$$ 26244.0i 1.03159i
$$866$$ − 1436.00i − 0.0563479i
$$867$$ 12039.0 0.471587
$$868$$ −13120.0 −0.513044
$$869$$ − 26304.0i − 1.02681i
$$870$$ −10152.0 −0.395615
$$871$$ 0 0
$$872$$ −14672.0 −0.569790
$$873$$ 15534.0i 0.602229i
$$874$$ 2304.00 0.0891693
$$875$$ 25680.0 0.992163
$$876$$ 2616.00i 0.100898i
$$877$$ 26314.0i 1.01318i 0.862186 + 0.506591i $$0.169095\pi$$
−0.862186 + 0.506591i $$0.830905\pi$$
$$878$$ 17968.0i 0.690650i
$$879$$ − 14994.0i − 0.575353i
$$880$$ 2304.00 0.0882589
$$881$$ −37506.0 −1.43429 −0.717145 0.696924i $$-0.754551\pi$$
−0.717145 + 0.696924i $$0.754551\pi$$
$$882$$ 1026.00i 0.0391692i
$$883$$ 6388.00 0.243458 0.121729 0.992563i $$-0.461156\pi$$
0.121729 + 0.992563i $$0.461156\pi$$
$$884$$ 0 0
$$885$$ −2160.00 −0.0820425
$$886$$ − 5208.00i − 0.197479i
$$887$$ −5472.00 −0.207138 −0.103569 0.994622i $$-0.533026\pi$$
−0.103569 + 0.994622i $$0.533026\pi$$
$$888$$ 2640.00 0.0997664
$$889$$ − 42880.0i − 1.61772i
$$890$$ − 2520.00i − 0.0949108i
$$891$$ 1944.00i 0.0730937i
$$892$$ − 1072.00i − 0.0402390i
$$893$$ 3264.00 0.122313
$$894$$ −10620.0 −0.397300
$$895$$ 72.0000i 0.00268904i
$$896$$ −2560.00 −0.0954504
$$897$$ 0 0
$$898$$ −26412.0 −0.981492
$$899$$ 46248.0i 1.71575i
$$900$$ −3204.00 −0.118667
$$901$$ −22140.0 −0.818635
$$902$$ 6048.00i 0.223255i
$$903$$ 9840.00i 0.362630i
$$904$$ − 2928.00i − 0.107725i
$$905$$ 28308.0i 1.03977i
$$906$$ 5928.00 0.217378
$$907$$ 7180.00 0.262853 0.131427 0.991326i $$-0.458044\pi$$
0.131427 + 0.991326i $$0.458044\pi$$
$$908$$ 7200.00i 0.263150i
$$909$$ −7182.00 −0.262059
$$910$$ 0 0
$$911$$ 27624.0 1.00464 0.502318 0.864683i $$-0.332481\pi$$
0.502318 + 0.864683i $$0.332481\pi$$
$$912$$ − 768.000i − 0.0278849i
$$913$$ 13248.0 0.480224
$$914$$ −16852.0 −0.609863
$$915$$ 11052.0i 0.399309i
$$916$$ − 11960.0i − 0.431408i
$$917$$ − 54960.0i − 1.97921i
$$918$$ 1620.00i 0.0582440i
$$919$$ −30256.0 −1.08602 −0.543011 0.839726i $$-0.682716\pi$$
−0.543011 + 0.839726i $$0.682716\pi$$
$$920$$ 3456.00 0.123849
$$921$$ − 20472.0i − 0.732438i
$$922$$ −33372.0 −1.19203
$$923$$ 0 0
$$924$$ −5760.00 −0.205076
$$925$$ 9790.00i 0.347993i
$$926$$ 31864.0 1.13079
$$927$$ 4680.00 0.165816
$$928$$ 9024.00i 0.319210i
$$929$$ 1926.00i 0.0680194i 0.999422 + 0.0340097i $$0.0108277\pi$$
−0.999422 + 0.0340097i $$0.989172\pi$$
$$930$$ − 5904.00i − 0.208172i
$$931$$ − 912.000i − 0.0321048i
$$932$$ 11304.0 0.397291
$$933$$ −26280.0 −0.922153
$$934$$ 37080.0i 1.29903i
$$935$$ 4320.00 0.151101
$$936$$ 0 0
$$937$$ 3962.00 0.138135 0.0690677 0.997612i $$-0.477998\pi$$
0.0690677 + 0.997612i $$0.477998\pi$$
$$938$$ − 33920.0i − 1.18073i
$$939$$ −11886.0 −0.413083
$$940$$ 4896.00 0.169883
$$941$$ 1074.00i 0.0372066i 0.999827 + 0.0186033i $$0.00592195\pi$$
−0.999827 + 0.0186033i $$0.994078\pi$$
$$942$$ 1956.00i 0.0676538i
$$943$$ 9072.00i 0.313282i
$$944$$ 1920.00i 0.0661978i
$$945$$ −3240.00 −0.111531
$$946$$ −7872.00 −0.270551
$$947$$ 4848.00i 0.166356i 0.996535 + 0.0831778i $$0.0265070\pi$$
−0.996535 + 0.0831778i $$0.973493\pi$$
$$948$$ −13152.0 −0.450588
$$949$$ 0 0
$$950$$ 2848.00 0.0972645
$$951$$ 21258.0i 0.724856i
$$952$$ −4800.00 −0.163413
$$953$$ −762.000 −0.0259009 −0.0129505 0.999916i $$-0.504122\pi$$
−0.0129505 + 0.999916i $$0.504122\pi$$
$$954$$ 13284.0i 0.450823i
$$955$$ 8208.00i 0.278120i
$$956$$ − 7248.00i − 0.245206i
$$957$$ 20304.0i 0.685826i
$$958$$ 12360.0 0.416841
$$959$$ −55080.0 −1.85467
$$960$$ − 1152.00i − 0.0387298i
$$961$$ 2895.00 0.0971770
$$962$$ 0 0
$$963$$ 108.000 0.00361397
$$964$$ 6328.00i 0.211422i
$$965$$ −19860.0 −0.662504
$$966$$ −8640.00 −0.287772
$$967$$ − 35804.0i − 1.19067i −0.803477 0.595336i $$-0.797019\pi$$
0.803477 0.595336i $$-0.202981\pi$$
$$968$$ 6040.00i 0.200551i
$$969$$ − 1440.00i − 0.0477394i
$$970$$ − 20712.0i − 0.685590i
$$971$$ −4260.00 −0.140793 −0.0703964 0.997519i $$-0.522426\pi$$
−0.0703964 + 0.997519i $$0.522426\pi$$
$$972$$ 972.000 0.0320750
$$973$$ 45040.0i 1.48398i
$$974$$ −23512.0 −0.773484
$$975$$ 0 0
$$976$$ 9824.00 0.322191
$$977$$ 28710.0i 0.940137i 0.882630 + 0.470069i $$0.155771\pi$$
−0.882630 + 0.470069i $$0.844229\pi$$
$$978$$ −8976.00 −0.293477
$$979$$ −5040.00 −0.164534
$$980$$ − 1368.00i − 0.0445910i
$$981$$ 16506.0i 0.537203i
$$982$$ 3816.00i 0.124006i
$$983$$ − 49524.0i − 1.60689i −0.595381 0.803444i $$-0.702999\pi$$
0.595381 0.803444i $$-0.297001\pi$$
$$984$$ 3024.00 0.0979691
$$985$$ −18756.0 −0.606717
$$986$$ 16920.0i 0.546493i
$$987$$ −12240.0 −0.394735
$$988$$ 0 0
$$989$$ −11808.0 −0.379649
$$990$$ − 2592.00i − 0.0832113i
$$991$$ 44408.0 1.42348 0.711739 0.702444i $$-0.247908\pi$$
0.711739 + 0.702444i $$0.247908\pi$$
$$992$$ −5248.00 −0.167968
$$993$$ − 27048.0i − 0.864393i
$$994$$ − 5280.00i − 0.168482i
$$995$$ 27984.0i 0.891610i
$$996$$ − 6624.00i − 0.210732i
$$997$$ 18398.0 0.584424 0.292212 0.956354i $$-0.405609\pi$$
0.292212 + 0.956354i $$0.405609\pi$$
$$998$$ 17888.0 0.567369
$$999$$ − 2970.00i − 0.0940607i
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1014.4.b.c.337.1 2
13.5 odd 4 1014.4.a.b.1.1 1
13.8 odd 4 78.4.a.e.1.1 1
13.12 even 2 inner 1014.4.b.c.337.2 2
39.8 even 4 234.4.a.b.1.1 1
52.47 even 4 624.4.a.i.1.1 1
65.34 odd 4 1950.4.a.c.1.1 1
104.21 odd 4 2496.4.a.k.1.1 1
104.99 even 4 2496.4.a.b.1.1 1
156.47 odd 4 1872.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.e.1.1 1 13.8 odd 4
234.4.a.b.1.1 1 39.8 even 4
624.4.a.i.1.1 1 52.47 even 4
1014.4.a.b.1.1 1 13.5 odd 4
1014.4.b.c.337.1 2 1.1 even 1 trivial
1014.4.b.c.337.2 2 13.12 even 2 inner
1872.4.a.e.1.1 1 156.47 odd 4
1950.4.a.c.1.1 1 65.34 odd 4
2496.4.a.b.1.1 1 104.99 even 4
2496.4.a.k.1.1 1 104.21 odd 4